Neural Control Variates with Automatic Integration
This addresses the problem of high variance in Monte Carlo integration for researchers and practitioners in computational fields, offering a novel but incremental improvement over existing neural network-based control variate methods.
The paper tackles the challenge of constructing expressive control variates with known integrals for Monte Carlo integration by proposing a method that uses neural networks to approximate the anti-derivative of the integrand, enabling automatic differentiation to create functions that reduce variance; results show it achieves lower variance than other methods in solving partial differential equations.
This paper presents a method to leverage arbitrary neural network architecture for control variates. Control variates are crucial in reducing the variance of Monte Carlo integration, but they hinge on finding a function that both correlates with the integrand and has a known analytical integral. Traditional approaches rely on heuristics to choose this function, which might not be expressive enough to correlate well with the integrand. Recent research alleviates this issue by modeling the integrands with a learnable parametric model, such as a neural network. However, the challenge remains in creating an expressive parametric model with a known analytical integral. This paper proposes a novel approach to construct learnable parametric control variates functions from arbitrary neural network architectures. Instead of using a network to approximate the integrand directly, we employ the network to approximate the anti-derivative of the integrand. This allows us to use automatic differentiation to create a function whose integration can be constructed by the antiderivative network. We apply our method to solve partial differential equations using the Walk-on-sphere algorithm. Our results indicate that this approach is unbiased and uses various network architectures to achieve lower variance than other control variate methods.